Data incompleteness--in the form of missingness, conesoring, rounding or grouping--is ubiquitous in biomedical research. When the degree of incompleteness is not associated with the values of the underlying, potentially incomplete data--a condition that we call ignorability--it is possible to obtain correct inferences without modeling the incompleteness mechanisms. When ignorability does not hold, or is not known to hold, inferences that do not model the incompleteness mechanism are potentially invaled, and to obtain correct inference one must model the incompleteness mechanism. This kind of modelingis typically computationally demanding, and results can be exquisitely sensitive to untestable assumptions, and conswquently we want to avoid it fi we can. We know that in some situations-- for example, when only a few data points are missing or incomplete--the effect of nonignorability cannot be large, and the incompleteness is presumably ignoragle for all pratical purposes. It would be useful to have a general, easily computed diagnostic that charaterizes data sets with respect to their potential for sensitivity to nonignorability. We have developed a diagnostic that measures the effect of small perturbations from ignorability on coefficient estimates in the univariate linear model with missing observations. In this application we propose to extens our analysis in a number of directions: i) We will develop a general class of diagnostics for Bayes and direct- likelihood inferences, and demonstrate its application to a number of important special cases. ii) We will develop an analogous theory for sensitivity to nonignorability in frequentist estimation and testing. iii) We will develop a general form of the diagnostic for the coarse-date model, a generalization of missing data that includes censoring and rounding as special cases. iv) We will analyze a number of real- world data sets that represent important cases where nonignorability is of interest, including dropout in longitudinal data, censored survival data, and cross-over in clinical trials.